 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;

theorem Th7:
  for A, J, f st not f in sqrt J holds J in Ideals(A,J,f)
   proof
   let A,J,f;
   assume
A1: not f in sqrt J;
    set I = J;
    I /\ multClSet(J,f) = {}
    proof
      assume I /\ multClSet(J,f) <> {}; then
      consider x be object such that
A4:   x in I /\ multClSet(J,f) by XBOOLE_0:def 1;
      x in I & x in multClSet(J,f) by A4,XBOOLE_0:def 4; then
      consider n1 be Nat such that
A5:   x = f|^n1;
      reconsider n1 as Element of NAT by ORDINAL1:def 12;
      sqrt J = {a where a is Element of A:
               ex n being Element of NAT st a|^n in J} by IDEAL_1:def 24;
      then not f|^n1 in J by A1;
      hence contradiction by A4,A5,XBOOLE_0:def 4;
    end;
    hence thesis;
end;
